# The archetype-genome exemplar in molecular dynamics and continuum mechanics

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## Abstract

We argue that mechanics and physics of solids rely on a fundamental exemplar: the apparent properties of a system depend on the building blocks that comprise it. Building blocks are referred to as archetypes and apparent system properties as the system genome. Three entities are of importance: the archetype properties, the conformation of archetypes, and the properties of interactions activated by that conformation. The combination of these entities into the system genome is called assembly. To show the utility of the archetype-genome exemplar, this work presents the mathematical ingredients and computational implementation of theories in solid mechanics that are (1) molecular and (2) continuum manifestations of the assembly process. Both coarse-grained molecular dynamics (CGMD) and the archetype-blending continuum (ABC) theories are formulated then applied to polymer nanocomposites (PNCs) to demonstrate the impact the components of the assembly triplet have on a material genome. CGMD simulations demonstrate the sensitivity of nanocomposite viscosities and diffusion coefficients to polymer chain types (archetype), polymer–nanoparticle interaction potentials (interaction), and the structural configuration (conformation) of dispersed nanoparticles. ABC simulations show the contributions of bulk polymer (archetype) properties, occluded region of bound rubber (interaction) properties, and microstructural binary images (conformation) to predictions of linear damping properties, the Payne effect, and localization/size effects in the same class of PNC material. The paper is light on mathematics. Instead, the focus is on the usefulness of the archetype-genome exemplar to predict system behavior inaccessible to classical theories by transitioning mechanics away from heuristic laws to mechanism-based ones. There are two core contributions of this research: (1) presentation of a fundamental axiom—the archetype-genome exemplar—to guide theory development in computational mechanics, and (2) demonstrations of its utility in modern theoretical realms: CGMD, and generalized continuum mechanics.

## Keywords

Materials genome Archetype Coarse-graining Continuum mechanics Molecular dynamics Polymer## Notes

### Acknowledgments

This work is supported by NSF CMMI Grants 0823327 and 0928320, as well as by NSF IDR CMM Grant I 1130948. M. Steven Greene warmly thanks the National Science Foundation GRFP for its support. Y. Li acknowledges financial support from Ryan Fellowship and Royal E. Cabell Terminal Year Fellowship at Northwestern University. W.K. Liu was also supported by the World Class University Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology. The authors would also like to thank Assistant Professor Khalil Elkhodary at the American University in Cairo, Egypt and Professor Shan Tang at Chongqing University in China for their challenging discussions and assistance with clarifying many of the concepts presented herein.

## References

- 1.ABAQUS theory manual (2011) Version 6.11Google Scholar
- 2.Abberton BC, Liu WK, Keten S (2013) Coarse-grained simulation of molecular mechanisms of recovery in thermally activated shape-memory polymers. J Mech Phys Solids. doi: 10.1016/j.jmps.2013.08.003
- 3.Accelrys (NASDAQ:ACCL) (2012) Materials studio. Available online at http://accelrys.com/products/materials-studio/. Accessed 18 Oct 2013
- 4.Akutagawa K, Yamaguchi K, Yamamoto A, Heguri H (2008) Mesoscopic mechanical analysis of filled elastomer with 3d-finite element analysis and transmission electron microtomography. Rubber Chem Technol 81:182–189Google Scholar
- 5.Askes H, Metrikine AV, Pichugin AV, Bennett T (2008) Four simplified gradient elasticity models for the simulation of dispersive wave propagation. Philos Mag 88(28–29):3415–3443Google Scholar
- 6.Bažant ZP, Jirásek M (2002) Nonlocal integral formulations of plasticity and damage: survey of progress. J Eng Mech 128(11):1119Google Scholar
- 7.Bažant ZP, Planas J (1998) Fracture and size effect in concrete and other quasibrittle materials. CRC Press, Boca RatonGoogle Scholar
- 8.Belytschko T, Mullen R (1978) On dispersive properties of finite element solutions. In: Achenbach J, Miklowitz J (eds) Modern problems in wave propagation. Wiley, New York, pp 67–82Google Scholar
- 9.Brini E, Algaer EA, Ganguly P, Li C, Rodríguez-Ropero F, van der Vegt NF (2013) Systematic coarse-graining methods for soft matter simulations: a review. Soft Matter 9(7):2108–2119Google Scholar
- 10.Brinson HF, Brinson LC (2008) Polymer engineering science and viscoelasticity: an introduction. Springer, New YorkGoogle Scholar
- 11.Brinson LC, Schmidt I, Lammering R (2004) Stress-induced transformation behavior of a polycrystalline niti shape memory alloy: micro and macromechanical investigations via in situ optical microscopy. J Mech Phys Solids 52(7):1549–1571MATHGoogle Scholar
- 12.Clifton T, Ferreira P (2013) Does dark energy really exist? Sci Am 58–65 (special edition: Extreme physics, probing the mysteries of the cosmos)Google Scholar
- 13.Cottrell JA, Hughes TJR, Bazilevs Y (2009) Isogeometric analysis: toward integration of CAD and FEA. Wiley, West SussexGoogle Scholar
- 14.Deng H, Liu Y, Gai D, Dikin DA, Putz KW, Chen W, Brinson LC, Burkhart C, Poldneff M, Jiang B, Papakonstantopoulos GJ (2012) Utilizing real and statistically reconstructed microstructures for the viscoelastic modeling of polymer nanocomposites. Compos Sci Technol 72(14):1725–1732Google Scholar
- 15.Dill K (2002) Molecular driving forces: statistical thermodynamics in chemistry & biology. Garland Science, New YorkGoogle Scholar
- 16.Doi M, Edwards S (1988) The theory of polymer dynamics, vol 73. Oxford University Press, New YorkGoogle Scholar
- 17.Dupres S, Long DR, Albouy PA, Sotta P (2009) Local deformation in carbon black-filled polyisoprene rubbers studied by nmr and X-ray diffraction. Macromolecules 42(7):2634–2644Google Scholar
- 18.Elkhodary KI, Greene MS, Tang S, Belytschko T, Liu WK (2013a) Archetype blending continuum theory. Comput Methods Appl Mech Eng 254:309–333MathSciNetGoogle Scholar
- 19.Elkhodary KI, Tang S, Liu WK (2013b) Inclusion clusters in the archetype-blending continuum theory. In: Handbook of micromechanics and nanomechanics. Pan Stanford Publishing, SingaporeGoogle Scholar
- 20.Eringen AC (1999) Microcontinuum field theories I: foundation and solids. Springer, New YorkGoogle Scholar
- 21.Eringen AC, Suhubi ES (1964) Nonlinear theory of simple microelastic solids. Int J Eng Sci 2(189–203):389–404MathSciNetGoogle Scholar
- 22.Faller R (2004) Automatic coarse graining of polymers. Polymer 45(11):3869–3876Google Scholar
- 23.Fish J, Kuznetsov S (2010) Computational continua. Int J Numer Methods Eng 84(7):774–802MATHMathSciNetGoogle Scholar
- 24.Fleck NA, Hutchinson JW (1997) Strain gradient plasticity. Adv Appl Mech 33:295–361Google Scholar
- 25.Fleck NA, Muller GM, Ashby MF, Hutchinson JW (1994) Strain gradient plasticity: theory and experiment. Acta Metall Mater 42(2):475–487Google Scholar
- 26.Forester T, Smith W (1998) Shake, rattle, and roll: efficient constraint algorithms for linked rigid bodies. J Comput Chem 19(1):102–111Google Scholar
- 27.Forrest JA, Dalnoki-Veress K, Stevens JR, Dutcher JR (1996) Effect of free surfaces on the glass transition temperature of thin polymer films. Phys Rev Lett 77(10):2002–2005Google Scholar
- 28.Frenkel D, Smit B (2001) Understanding molecular simulation: from algorithms to applications, vol 1. Academic press, New YorkGoogle Scholar
- 29.Fröhlich J, Niedermeier W, Luginsland HD (2005) The effect of filler–filler and filler–elastomer interaction on rubber reinforcement. Compos Part A Appl S 36(4):449–460Google Scholar
- 30.Gao H, Huang Y, Nix WD, Hutchinson JW (1999) Mechanism-based strain gradient plasticity: i. theory. J Mech Phys Solids 47(6):1239–1263MATHMathSciNetGoogle Scholar
- 31.Geers MGD, Kouznetsova VG, Brekelmans WAM (2010) Multi-scale computational homogenization: trends and challenges. J Comput Appl Math 234(7):2175–2182MATHGoogle Scholar
- 32.Genomic Science Program (2011) About the human genome project. Available online at http://web.ornl.gov/sci/techresources/Human_Genome/project/index.shtml. Accessed 8 Oct 2013
- 33.Germain P (1973) The method of virtual power in continuum mechanics. Part 2: microstructure. SIAM J Appl Math 25(3):556–575MATHMathSciNetGoogle Scholar
- 34.Ghanem R, Spanos PD (1991) Stochastic finite elements: a spectral approach. Springer, New YorkMATHGoogle Scholar
- 35.Gibson LJ, Ashby MF (1997) Cellular solids: structure and properties, 2nd edn. Cambridge University Press, New YorkGoogle Scholar
- 36.Gonella S, Greene MS, Liu WK (2011) Characterization of heterogeneous solids via wave methods in computational microelasticity. J Mech Phys Solids 59(5):959–974MATHGoogle Scholar
- 37.Gonzalez J, Knauss WG (1998) Strain inhomogeneity and discontinuous crack growth in a particulate composite. J Mech Phys Solids 46(10):1981–1995MATHGoogle Scholar
- 38.Greene MS, Liu Y, Chen W, Liu WK (2011) Computational uncertainty analysis in multiresolution materials via stochastic constitutive theory. Comput Methods Appl Mech Eng 200(1–4):309–325MathSciNetGoogle Scholar
- 39.Greene MS, Gonella S, Liu WK (2012) Microelastic wave field signatures and their implications for microstructure identification. Int J Solids Struct 49(22):3148–3157Google Scholar
- 40.Greene MS, Xu H, Tang S, Chen W, Liu WK (2013) A generalized uncertainty propagation criterion from benchmark studies of microstructured material systems. Comput Methods Appl Mech Eng 254:271–291MathSciNetGoogle Scholar
- 41.Greer JR, Oliver WC, Nix WD (2005) Size dependence of mechanical properties of gold at the micron scale in the absence of strain gradients. Acta Mater 53(6):1821–1830Google Scholar
- 42.Gurtin ME, Anand L (2005) A theory of strain-gradient plasticity for isotropic, plastically irrotational materials. Part i: small deformations. J Mech Phys Solids 53(7):1624–1649MATHMathSciNetGoogle Scholar
- 43.Hao S, Liu WK, Qian D (2000) Localization-induced band and cohesive model. J Appl Mech 67(4):803–812MATHGoogle Scholar
- 44.Harmandaris V, Kremer K (2009) Dynamics of polystyrene melts through hierarchical multiscale simulations. Macromolecules 42(3):791–802Google Scholar
- 45.Hoogerbrugge P, Koelman J (1992) Simulating microscopic hydrodynamic phenomena with dissipative particle dynamics. Europhys Lett 19(3):155Google Scholar
- 46.Hoover W (1985) Canonical dynamics: equilibrium phase-space distributions. Phys Rev A 31(3):1695Google Scholar
- 47.Hua CC, Schieber JD (1998) Segment connectivity, chain-length breathing, segmental stretch, and constraint release in reptation models. i. Theory and single-step strain predictions. J Chem Phys 109:10018–10027Google Scholar
- 48.Hughes T, Cottrell J, Bazilevs Y (2005) Isogeometric analysis: cad, finite elements, nurbs, exact geometry and mesh refinement. Comput Methods Appl Mech Eng 194(39–41):4135–4195MATHMathSciNetGoogle Scholar
- 49.Hwang KC, Jiang H, Huang Y, Gao H, Hu N (2002) A finite deformation theory of strain gradient plasticity. J Mech Phys Solids 50(1):81–99MATHMathSciNetGoogle Scholar
- 50.Jensen MK, Khaliullin R, Schieber JD (2012) Self-consistent modeling of entangled network strands and linear dangling structures in a single-strand mean-field slip-link model. Rheol Acta 51(1):21–35Google Scholar
- 51.Jordan J, Jacob KI, Tannenbaum R, Sharaf MA, Jasiuk I (2005) Experimental trends in polymer nanocomposites: a review. Mater Sci Eng A Struct 393:1–11 (review article)Google Scholar
- 52.Kamberaj H, Low R, Neal M (2005) Time reversible and symplectic integrators for molecular dynamics simulations of rigid molecules. J Chem Phys 122(224):114Google Scholar
- 53.Karásek L, Sumita M (1996) Characterization of dispersion state of filler and polymer–filler interactions in rubber–carbon black composites. J Mater Sci 31(2):281–289Google Scholar
- 54.Koiter WT (1964) Couple stresses in the theory of elasticity, i and ii. Proc K Ned Akad van Wet Ser B 67(1):17–44MATHGoogle Scholar
- 55.Kopacz A, Patankar N, Liu W (2012) The immersed molecular finite element method. Comput Methods Appl Mech Eng 233–236:28–39MathSciNetGoogle Scholar
- 56.Kremer K, Grest G (1990) Dynamics of entangled linear polymer melts: a molecular-dynamics simulation. J Chem Phys 92:5057Google Scholar
- 57.Kremer K, Müller-Plathe F (2002) Multiscale simulation in polymer science. Mol Simul 28(8–9):729–750Google Scholar
- 58.Kröger M (1999) Efficient hybrid algorithm for the dynamic creation of wormlike chains in solutions, brushes, melts and glasses. Comput Phys Commun 118(2):278–298Google Scholar
- 59.Kröger M (2004) Simple models for complex nonequilibrium fluids. Phys Rep 390(6):453–551MathSciNetGoogle Scholar
- 60.Kröger M, Hess S (2000) Rheological evidence for a dynamical crossover in polymer melts via nonequilibrium molecular dynamics. Phys Rev Lett 85(5):1128–1131Google Scholar
- 61.Lakes R (1993) Materials with structural hierarhcy. Nature 361(6412):511–514Google Scholar
- 62.Lakes RS (1999) Viscoelastic solids. CRC Press, Boca RatonGoogle Scholar
- 63.Leblanc JL (2000) Elastomer–filler interactions and the rheology of filled rubber compounds. J Appl Polym Sci 78(8):1541– 1550Google Scholar
- 64.Li Y, Kröger M, Liu WK (2011) Primitive chain network study on uncrosslinked and crosslinked \(cis\)-polyisoprene polymers. Polymer 52(25):5867–5878Google Scholar
- 65.Li Y, Kröger M, Liu WK (2012a) Nanoparticle effect on the dynamics of polymer chains and their entanglement network. Phys Rev Lett 109(11):118,001Google Scholar
- 66.Li Y, Kröger M, Liu WK (2012b) Nanoparticle geometrical effect on structure, dynamics and anisotropic viscosity of polyethylene nanocomposites. Macromolecules 45(4):2099–2112Google Scholar
- 67.Li Y, Tang S, Abberton B, Kröger M, Burkhart C, Jiang B, Papakonstantopoulos G, Poldneff M, Liu WK (2012c) A predictive multiscale computational framework for viscoelastic properties of linear polymers. Polymer 53(25):5935–5952Google Scholar
- 68.Li Y, Abberton B, Kröger M, Liu WK (2013) Challenges in multiscale modeling of polymer dynamics. Polymers 5(2):751– 832Google Scholar
- 69.Litvinov VM, Orza RA, Klüppel M, van Duin M, Magusin PCMM (2011) Rubber–filler interactions and network structure in relation to stress–strain behavior of vulcanized, carbon black filled epdm. Macromolecules 44(12):4887–4900Google Scholar
- 70.Liu WK, Belytschko T, Mani A (1986a) Probabilistic finite elements for nonlinear structural dynamics. Comput Methods Appl Mech Eng 56(1):61–81MATHGoogle Scholar
- 71.Liu WK, Belytschko T, Mani A (1986b) Random field finite elements. Int J Numer Methods Eng 23(10):1831–1845MATHMathSciNetGoogle Scholar
- 72.Liu WK, Karpov EG, Zhang S, Park H (2004) An introduction to computational nanomechanics and materials. Comput Methods Appl Mech Eng 193(17):1529–1578MATHMathSciNetGoogle Scholar
- 73.Liu WK, Karpov E, Park H, Wiley J (2006) Nano mechanics and materials: theory, multiscale methods and applications. Wiley, New YorkGoogle Scholar
- 74.Liu Y, Greene MS, Chen W, Dikin DA, Liu WK (2012) Computational microstructure characterization and reconstruction for stochastic multiscale material design. Comput Aided Des 45(1):65–76Google Scholar
- 75.Malvern LE (1969) Introduction to the mechanics of a continuous medium. Prentice Hall, Englewood CliffsGoogle Scholar
- 76.Martyna G, Klein M, Tuckerman M (1992) Nosé-hoover chains: the canonical ensemble via continuous dynamics. J Chem Phys 97:2635Google Scholar
- 77.Matsen M (2001) The standard gaussian model for block copolymer melts. J Phys Condens Matter 14(2):R21Google Scholar
- 78.McVeigh C, Liu WK (2008) Linking microstructure and properties through a predictive multiresolution continuum. Comput Methods Appl Mech Eng 197(41–42):3268–3290MATHMathSciNetGoogle Scholar
- 79.McVeigh C, Liu WK (2010) Multiresolution continuum modeling of micro-void assisted dynamic adiabatic shear band propagation. J Mech Phys Solids 58(2):187–205MATHMathSciNetGoogle Scholar
- 80.McVeigh C, Vernerey F, Liu WK, Moran B, Olson G (2007) An interactive micro-void shear localization mechanism in high strength steels. J Mech Phys Solids 55(2):225–244Google Scholar
- 81.Milano G, Goudeau S, Müller-Plathe F (2005) Multicentered gaussian-based potentials for coarse-grained polymer simulations: linking atomistic and mesoscopic scales. J Polym Sci Part B 43(8):871–885Google Scholar
- 82.Miller RE, Tadmor EB (2009) A unified framework and performance benchmark of fourteen multiscale atomistic/continuum coupling methods. Model Simul Mater Sci 17(5):053,001Google Scholar
- 83.Mindlin RD (1964) Micro-structure in linear elasticity. Arch Ration Mech Anal 16(1):51–78MATHMathSciNetGoogle Scholar
- 84.Mindlin RD (1965) Second gradient of strain and surface-tension in linear elasticity. Int J Solids Struct 1(4):417–438Google Scholar
- 85.Müller-Plathe F (2002) Scale-hopping in computer simulations of polymers. Soft Mater 1(1):1–31Google Scholar
- 86.Müller-Plathe F (2012) Ibisco:it is boltzmann inversion software for coarse graining simulations. Available online at http://www.theo.chemie.tu-darmstadt.de/ibisco/IBISCO.html. Accessed 18 Oct 2013
- 87.Mura T (1987) Micromechanics of defects in solids mechanics of elastic and inelastic solids, 2nd edn. Kluwer Academic, DordrechtGoogle Scholar
- 88.Naraghi M, Arshad SN, Chasiotis I (2011) Molecular orientation and mechanical property size effects in electrospun polyacrylonitrile nanofibers. Polymer 52(7):1612–1618Google Scholar
- 89.National Science and Technology Council (2011) Materials genome initiative for global competitiveness. Tech. rep, Office of Science and Technology PolicyGoogle Scholar
- 90.Nemat-Nasser S, Hori M (1999) Micromechanics: ovrall properties of heterogeneous materials. Elsevier, New YorkGoogle Scholar
- 91.Nix WD, Gao H (1998) Indentation size effects in crystalline materials: a law for strain gradient plasticity. J Mech Phys Solids 46(3):411–425MATHGoogle Scholar
- 92.Noid W, Chu J, Ayton G, Krishna V, Izvekov S, Voth G, Das A, Andersen H (2008a) The multiscale coarse-graining method. i. A rigorous bridge between atomistic and coarse-grained models. J Chem Phys 128(24):244,114Google Scholar
- 93.Noid W, Liu P, Wang Y, Chu J, Ayton G, Izvekov S, Andersen H, Voth G (2008b) The multiscale coarse-graining method. ii. Numerical implementation for coarse-grained molecular models. J Chem Phys 128(24):244,115Google Scholar
- 94.Oden JT, Prudhomme S (2011) Control of modeling error in calibration and validation processes for predictive stochastic models. Int J Numer Methods Eng 87(1–5):262–272MATHGoogle Scholar
- 95.Olson GB (2000) Designing a new material world. Science 288:993–998Google Scholar
- 96.Ostoja-Starzewski M (1998) Random field models of heterogeneous materials. Int J Solids Struct 35(19):2429–2455MATHGoogle Scholar
- 97.Ostoja-Starzewski M (2006) Material spatial randomness: from statistical to representative volume element. Probab Eng Mech 21(2):112–132Google Scholar
- 98.Padding J, Briels W (2011) Systematic coarse-graining of the dynamics of entangled polymer melts: the road from chemistry to rheology. J Phys Condens Matter 23(23):233,101Google Scholar
- 99.Papakonstantopoulos G, Doxastakis M, Nealey P, Barrat J, de Pablo J (2007) Calculation of local mechanical properties of filled polymers. Phys Rev E 75(3):031,803Google Scholar
- 100.Papargyri-Beskou S, Polyzos D, Beskos DE (2009) Wave dispersion in gradient elastic solids and structures: a unified treatment. Int J Solids Struct 46(21):3751–3759MATHGoogle Scholar
- 101.Park H, Karpov E, Liu WK (2004) A temperature equation for coupled atomistic/continuum simulations. Comput Methods Appl Mech Eng 193(17):1713–1732MATHMathSciNetGoogle Scholar
- 102.Parrinello M, Rahman A (1981) Polymorphic transitions in single crystals: a new molecular dynamics method. J Appl Phys 52(12):7182–7190Google Scholar
- 103.Paul W, Smith GD (2004) Structure and dynamics of amorphous polymers: computer simulations compared to experiment and theory. Rep Prog Phys 67(7):1117Google Scholar
- 104.Plimpton S et al (1995) Fast parallel algorithms for short-range molecular dynamics. J Comput Phys 117(1):1–19MATHGoogle Scholar
- 105.Qian D, Wagner GJ, Liu WK (2004) A multiscale projection method for the analysis of carbon nanotubes. Comput Methods Appl Mech Eng 193(17):1603–1632MATHGoogle Scholar
- 106.Qiao R, Deng H, Putz KW, Brinson LC (2011) Effect of particle agglomeration and interphase on the glass transition temperature of polymer nanocomposites. J Polym Sci Polym Phys 49(10):740–748Google Scholar
- 107.Reith D, Pütz M, Müller-Plathe F (2003) Deriving effective mesoscale potentials from atomistic simulations. J Comput Chem 24(13):1624–1636Google Scholar
- 108.Rho JY, Kuhn-Spearing L, Zioupos P (1998) Mechanical properties and the hierarchical structure of bone. Med Eng Phys 20:92–102Google Scholar
- 109.Rouse P Jr (1953) A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J Chem Phys 21:1272Google Scholar
- 110.Schieber JD, Neergaard J, Gupta S (2003) A full-chain, temporary network model with sliplinks, chain-length fluctuations, chain connectivity and chain stretching. J Rheol 47:213Google Scholar
- 111.Stankovich S, Dikin DA, Dommett GHB, Kohlhaas KM, Zimney EJ, Stach EA, Piner RD, Nguyen ST, Ruoff RS (2006) Graphene-based composite materials. Nature 442(7100):282–286Google Scholar
- 112.Starr F, Douglas J (2011) Modifying fragility and collective motion in polymer melts with nanoparticles. Phys Rev Lett 106(11):115,702Google Scholar
- 113.Stölken JS, Evans AG (1998) A microbend test method for measuring the plasticity length scale. Acta Mater 46(14):5109– 5115Google Scholar
- 114.Sun H (1998) Compass: An ab initio force-field optimized for condensed-phase applications overview with details on alkane and benzene compounds. J Phys Chem B 102(38):7338–7364Google Scholar
- 115.Szleifer I, Carignano M (2009) Tethered polymer layers. Adv Chem Phys 94:165–260Google Scholar
- 116.Tang S, Greene MS, Liu WK (2011) A variable constraint tube model for size effects in polymer nano-structures. Appl Phys Lett 99(191):910Google Scholar
- 117.Tang S, Greene MS, Liu WK (2012a) A renormalization approach to model interaction in microstructured solids: application to porous elastomer. Comput Methods Appl Mech Eng 217–220:213–225MathSciNetGoogle Scholar
- 118.Tang S, Greene MS, Liu WK (2012b) Two-scale mechanism-based theory of nonlinear viscoelasticity. J Mech Phys Solids 60(2):199–226MATHMathSciNetGoogle Scholar
- 119.Tang S, Kopacz AM, Chan S, Olson GB, Liu WK (2013a) Three-dimensional ductile fracture analysis with a hybrid multiresolution approach and microtomography. J Mech Phys Solids 61(11):2108–2124Google Scholar
- 120.Tang S, Kopacz AM, OKeeffe SC, Olson GB, Liu WK (2013b) Concurrent multiresolution finite element: formulation and algorithmic aspects. Comput Mech. doi: 10.1007/s00466-013-0874-3
- 121.Thurner PJ, Erickson B, Jungmann R, Schriock Z, Weaver JC, Fantner GE, Schitter G, Morse DE, Hansma PK (2007) High-speed photography of compressed human trabecular bone correlates whitening to microscopic damage. Eng Frac Mech 74(12):1928–1941Google Scholar
- 122.Tian R, Chan S, Tang S, Kopacz AM, Wang JS, Jou HJ, Siad L, Lindgren LE, Olson GB, Liu WK (2010) A multiresolution continuum simulation of the ductile fracture process. J Mech Phys Solids 58(10):1681–1700MATHGoogle Scholar
- 123.Ting CS, Sachse W (1978) Measurement of ultrasonic dispersion by phase comparison of continuous harmonic waves. J Acoust Soc Am 64(3):852–857Google Scholar
- 124.Torquato S (2002) Statistical description of microstructures. Annu Rev Mater Res 32:77–111Google Scholar
- 125.Toupin RA (1962) Elastic materials with couple-stresses. Arch Ration Mech Anal 11(1):385–414MATHMathSciNetGoogle Scholar
- 126.Tschöp W, Kremer K, Batoulis J, Bürger T, Hahn O (1998) Simulation of polymer melts. i. Coarse-graining procedure for polycarbonates. Acta Polym 49(2–3):61–74Google Scholar
- 127.Tuckerman M, Berne B, Martyna G (1992) Reversible multiple time scale molecular dynamics. J Chem Phys 97:1990Google Scholar
- 128.Uchic MD, Groeber MA, Dimiduk DM, Simmons JP (2006) 3d microstructural characterization of nickel superalloys via serial-sectioning using a dual beam fib-sem. Scripta Materialia 55(1):23–28 Google Scholar
- 129.Vacatello M (2001) Monte carlo simulations of polymer melts filled with solid nanoparticles. Macromolecules 34(6):1946–1952Google Scholar
- 130.Vernerey F, Liu WK, Moran B (2007) Multi-scale micromorphic theory for hierarchical materials. J Mech Phys Solids 55(12):2603–2651MATHMathSciNetGoogle Scholar
- 131.Vernerey F, Liu WK, Moran B, Olson G (2008) A micromorphic model for the multiple scale failure of heterogeneous materials. J Mech Phys Solids 56(4):1320–1347MATHMathSciNetGoogle Scholar
- 132.Vernerey FJ, Liu WK, Moran B, Olson G (2009) Multi-length scale micromorphic process zone model. Comput Mech 44(3):433–445MATHGoogle Scholar
- 133.Wagner G, Karpov E, Liu W (2004) Molecular dynamics boundary conditions for regular crystal lattices. Comput Methods Appl Mech Eng 193(17):1579–1601MATHMathSciNetGoogle Scholar
- 134.Wagner GJ, Liu WK (2003) Coupling of atomistic and continuum simulations using a bridging scale decomposition. J Comput Phys 190(1):249–274MATHGoogle Scholar
- 135.Wang L, Hu H (2005) Flexural wave propagation in single-walled carbon nanotubes. Phys Rev B 71(19):195,412Google Scholar
- 136.Xiao SP, Belytschko T (2004) A bridging domain method for coupling continua with molecular dynamics. Comput Methods Appl Mech Eng 193(17–20):1645–1669MATHMathSciNetGoogle Scholar
- 137.Xu H, Greene MS, Deng H, Dikin D, Brinson LC, Liu WK, Burkhart C, Papakonstantopoulos GJ, Poldneff M, Chen W (2013) Stochastic reassembly strategy for managing information complexity in heterogeneous materials analysis and design. J Mech Des (In press)Google Scholar
- 138.Yin D, Zhang Y, Peng Z, Zhang Y (2003) Effect of fillers and additives on the properties of sbr vulcanizates. J Appl Polym Sci 88(3):775–782Google Scholar
- 139.Yin X, Chen W, To A, McVeigh C, Liu WK (2008) Statistical volume element method for predicting microstructure-constitutive property relations. Comput Methods Appl Mech Eng 197(43–44):3516–3529MATHMathSciNetGoogle Scholar
- 140.Yurekli K, Krishnamoorti R, Tse MF, McElrath KO, Tsou AH, Wang HC (2001) Structure and dynamics of carbon black-filled elastomers. J Polym Sci Polym Phys 39(2):256–275Google Scholar